Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [187,2,Mod(4,187)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(187, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([4, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("187.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 187 = 11 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 187.p (of order \(20\), degree \(8\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(1.49320251780\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −1.64595 | + | 2.26545i | 0.433352 | + | 0.850501i | −1.80509 | − | 5.55550i | 0.526458 | − | 3.32392i | −2.64004 | − | 0.418141i | −0.503952 | + | 0.989061i | 10.2304 | + | 3.32405i | 1.22780 | − | 1.68992i | 6.66366 | + | 6.66366i |
4.2 | −1.38201 | + | 1.90217i | 1.08670 | + | 2.13277i | −1.09027 | − | 3.35550i | −0.583785 | + | 3.68587i | −5.55871 | − | 0.880413i | 0.909545 | − | 1.78508i | 3.41722 | + | 1.11032i | −1.60442 | + | 2.20830i | −6.20435 | − | 6.20435i |
4.3 | −1.31417 | + | 1.80880i | −0.639016 | − | 1.25414i | −0.926682 | − | 2.85203i | −0.322855 | + | 2.03843i | 3.10826 | + | 0.492300i | −1.13928 | + | 2.23597i | 2.12384 | + | 0.690076i | 0.598833 | − | 0.824223i | −3.26282 | − | 3.26282i |
4.4 | −1.04806 | + | 1.44253i | −0.804161 | − | 1.57826i | −0.364434 | − | 1.12161i | −0.116829 | + | 0.737628i | 3.11950 | + | 0.494080i | 1.20459 | − | 2.36415i | −1.39168 | − | 0.452186i | −0.0808598 | + | 0.111294i | −0.941609 | − | 0.941609i |
4.5 | −0.887944 | + | 1.22215i | −0.0267877 | − | 0.0525737i | −0.0871729 | − | 0.268291i | 0.507206 | − | 3.20238i | 0.0880390 | + | 0.0139440i | 1.63147 | − | 3.20194i | −2.46815 | − | 0.801951i | 1.76131 | − | 2.42423i | 3.46341 | + | 3.46341i |
4.6 | −0.830083 | + | 1.14251i | 0.831391 | + | 1.63170i | 0.00173994 | + | 0.00535500i | 0.0745546 | − | 0.470719i | −2.55435 | − | 0.404570i | −1.69465 | + | 3.32594i | −2.69377 | − | 0.875258i | −0.207866 | + | 0.286102i | 0.475915 | + | 0.475915i |
4.7 | −0.273504 | + | 0.376447i | 0.441712 | + | 0.866909i | 0.551127 | + | 1.69619i | −0.114620 | + | 0.723685i | −0.447155 | − | 0.0708224i | −0.487084 | + | 0.955955i | −1.67434 | − | 0.544026i | 1.20693 | − | 1.66120i | −0.241080 | − | 0.241080i |
4.8 | −0.212815 | + | 0.292914i | −1.13391 | − | 2.22542i | 0.577525 | + | 1.77744i | −0.417832 | + | 2.63808i | 0.893168 | + | 0.141464i | −1.09590 | + | 2.15082i | −1.33223 | − | 0.432866i | −1.90338 | + | 2.61978i | −0.683811 | − | 0.683811i |
4.9 | −0.0756214 | + | 0.104084i | 1.34406 | + | 2.63788i | 0.612919 | + | 1.88637i | 0.0370110 | − | 0.233679i | −0.376200 | − | 0.0595843i | 2.00593 | − | 3.93685i | −0.487406 | − | 0.158368i | −3.38852 | + | 4.66390i | 0.0215233 | + | 0.0215233i |
4.10 | 0.201795 | − | 0.277747i | −0.935793 | − | 1.83660i | 0.581612 | + | 1.79002i | 0.314547 | − | 1.98597i | −0.698947 | − | 0.110702i | 0.501695 | − | 0.984632i | 1.26756 | + | 0.411855i | −0.734024 | + | 1.01030i | −0.488124 | − | 0.488124i |
4.11 | 0.668874 | − | 0.920626i | −0.202818 | − | 0.398052i | 0.217874 | + | 0.670546i | −0.507205 | + | 3.20236i | −0.502117 | − | 0.0795275i | 2.13204 | − | 4.18437i | 2.92758 | + | 0.951227i | 1.64605 | − | 2.26559i | 2.60893 | + | 2.60893i |
4.12 | 0.765751 | − | 1.05397i | 1.05793 | + | 2.07631i | 0.0935641 | + | 0.287961i | 0.683608 | − | 4.31613i | 2.99848 | + | 0.474912i | −1.20165 | + | 2.35837i | 2.85317 | + | 0.927052i | −1.42849 | + | 1.96614i | −4.02558 | − | 4.02558i |
4.13 | 0.827467 | − | 1.13891i | 0.330414 | + | 0.648475i | 0.00561864 | + | 0.0172924i | −0.427401 | + | 2.69851i | 1.01196 | + | 0.160279i | −1.38515 | + | 2.71850i | 2.70208 | + | 0.877960i | 1.45201 | − | 1.99852i | 2.71970 | + | 2.71970i |
4.14 | 1.09096 | − | 1.50157i | −1.13445 | − | 2.22649i | −0.446504 | − | 1.37420i | 0.176770 | − | 1.11608i | −4.58088 | − | 0.725540i | −0.144965 | + | 0.284510i | 0.979837 | + | 0.318368i | −1.90691 | + | 2.62463i | −1.48304 | − | 1.48304i |
4.15 | 1.44335 | − | 1.98660i | −0.272661 | − | 0.535128i | −1.24529 | − | 3.83260i | 0.0688455 | − | 0.434673i | −1.45663 | − | 0.230707i | −1.33328 | + | 2.61671i | −4.74045 | − | 1.54026i | 1.55134 | − | 2.13523i | −0.764154 | − | 0.764154i |
4.16 | 1.49639 | − | 2.05960i | 1.05967 | + | 2.07972i | −1.38474 | − | 4.26180i | −0.119706 | + | 0.755793i | 5.86908 | + | 0.929571i | 0.600644 | − | 1.17883i | −6.00731 | − | 1.95189i | −1.43899 | + | 1.98060i | 1.37751 | + | 1.37751i |
38.1 | −2.30228 | + | 0.748055i | 1.85991 | − | 0.294580i | 3.12286 | − | 2.26889i | 1.21515 | + | 2.38487i | −4.06166 | + | 2.06952i | 0.345011 | + | 0.0546444i | −2.64667 | + | 3.64282i | 0.519302 | − | 0.168732i | −4.58163 | − | 4.58163i |
38.2 | −1.96554 | + | 0.638643i | −1.63407 | + | 0.258811i | 1.83745 | − | 1.33498i | 0.781962 | + | 1.53469i | 3.04654 | − | 1.55229i | 3.21586 | + | 0.509342i | −0.329459 | + | 0.453461i | −0.249973 | + | 0.0812212i | −2.51709 | − | 2.51709i |
38.3 | −1.93141 | + | 0.627553i | −2.69128 | + | 0.426258i | 1.71848 | − | 1.24855i | −0.435016 | − | 0.853767i | 4.93047 | − | 2.51220i | −4.37611 | − | 0.693108i | −0.148208 | + | 0.203990i | 4.20815 | − | 1.36731i | 1.37598 | + | 1.37598i |
38.4 | −1.61464 | + | 0.524629i | 0.410158 | − | 0.0649626i | 0.713800 | − | 0.518606i | −0.824518 | − | 1.61821i | −0.628176 | + | 0.320072i | −0.362922 | − | 0.0574812i | 1.11535 | − | 1.53515i | −2.68916 | + | 0.873761i | 2.18026 | + | 2.18026i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
17.c | even | 4 | 1 | inner |
187.p | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 187.2.p.a | ✓ | 128 |
11.c | even | 5 | 1 | inner | 187.2.p.a | ✓ | 128 |
17.c | even | 4 | 1 | inner | 187.2.p.a | ✓ | 128 |
187.p | even | 20 | 1 | inner | 187.2.p.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
187.2.p.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
187.2.p.a | ✓ | 128 | 11.c | even | 5 | 1 | inner |
187.2.p.a | ✓ | 128 | 17.c | even | 4 | 1 | inner |
187.2.p.a | ✓ | 128 | 187.p | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(187, [\chi])\).